Papers › Two-body collapse model for self-gravitating flow of dark matter and generalized...
Two-body collapse model for self-gravitating flow of dark matter and generalized stable clustering hypothesis for pairwise velocity
Zhijie Xu
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Analytical tools are extremely hard to find for non-linear gravitational collpase. Only a few simple but powerful tools exist so far. Two examples are the spherical collapse model (SCM) and stable clustering hypothesis (SCH). We present a new analytical tool, a two-body collapse model (TBCM), that plays the same fundamental role as harmonic oscillator in dynamics. For convenience, TBCM is formulated for gravity with any potential exponent n in a static background with a fixed damping (n=-1 for Newtonian gravity). The competition between gravity, expanding background (or damping), and angular momentum classifies two-body collapse into: 1) free fall collapse, where free fall time is greater if same system starts to collapse at earlier time; 2) equilibrium collapse that persists longer in time, whose perturbative solutions lead to power-law evolution of system energy and momentum. Two critical values βₛ₁=1 and βₛ₂=1/3π are identified that quantifies the competition between damping and gravity. Value βₛ₂ only exists for discrete values of potential exponent n=(2-6m)/(1+3m)= -1,-10/7... for integer m. Critical density ratio (Δ_c=18π²) is obtained for n=-1 that is consistent with SCM. TBCM predicts angular velocity ∝Hr^(-3/2) for two-body system of size r. The isothermal density is a result of extremely fast mass accretion. TBCM is able to demonstrate SCH, i.e. mean pairwise velocity (first moment) ⟨Δu⟩=-Hr. A generalized SCH is developed for higher order moments ⟨Δu²ᵐ⁺¹⟩=-(2m+1)⟨Δu²ᵐ⟩Hr that is validated by N-body simulation. Energy evolution in TBCM is independent of particle mass and energy equipartition does not apply. TBCM can be considered as a non-radial SCM. Both models predict the same critical density ratio, while TBCM contains much richer information.
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